Relative topos theory via stacks

I am glad to announce the first version of our joint work with Riccardo Zanfa on relative toposes:

Relative topos theory via stacks

Here is the abstract:

We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site (C,J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a 'petit' topos associated with a 'gros' topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.

We shall progressively release expanded versions of this text contaning new developments in the directions sketched in the introduction.

This work has been recently presented at the conference Toposes online:

The following video focuses on the specialization of the fundamental adjunction in the setting of presheaves (or discrete fibrations):

Video of my talk on relative toposes

The video of my yesteday talk on relative toposes is already available from YouTube:

Thanks again to the organizers for their invitation, and for making the recording available so quickly!

A talk on relative toposes

Next Tuesday I shall give a talk for ItaCa Fest 2021 on my work in progress with Riccardo Zanfa providing new foundations for relative topos theory based on stacks:

15
June2021
Relative topos theory via stacksIn this talk, based on joint work with Riccardo Zanfa, we shall introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of (relatively small) toposes over the topos of sheaves on a given site (C, J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘gros’ topos, thereby providing an answer to a problem posed by Grothendieck in the seventies.
14:30 (Italian time)ItaCa Fest 2021

Here is the Zoom link to attend the talk: https://zoom.us/j/94880770089?pwd=clgxK2VkVEE5Ymw5ME1QWktiWExUZz09

Thanks to the ItaCa Fest organizers for this invitation, and looking forward to seeing many of you there!